Questions tagged [parabolic-pde]
A partial differential equation that describes diffusion phenomena.
22
questions with no upvoted or accepted answers
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views
Form of nonlinear diffusion equation
Consider the following nonlinear diffusion problem,
$$
\frac{\partial u}{\partial t} = x^{-2}\frac{\partial}{\partial x}\left(x^2 u^4 \frac{\partial u}{\partial x}\right), \quad 0 < x < 1
$$
We ...
- 412
2
votes
0
answers
74
views
Numerical scheme for the heat equation on the icosahedral hexagonal grid
I have a predefined grid(like this) that is spawned from a regular icosahedradron. It consists of many hexagons and 12 pentagons (corresponding to icosahedradron vertexes). I can tweak the granularity ...
- 121
2
votes
0
answers
37
views
Convergent Finite Difference Scheme for Parabolic Equation
Consider the PDE $$u_t = b_{11}u_{xx} + 2b_{12}u_{xy} + b_{22}u_{yy},$$
where $b_{11}, b_{22} > 0$, and $b_{12}^2 < b_1b_2$.
In Strikwerda's book, the ADI scehme \begin{align*}
\left( 1 - \frac{...
- 141
2
votes
0
answers
77
views
Unusual boundary conditions on Matlab
I'm trying to solve the following PDE by Matlab,
$$
u_t-\Delta u = 0, \quad \text{in}\quad \Omega\times (0,T)
\tag{1}
$$
$$
u_t-\Delta_\Gamma u + \partial_\nu u=0,\quad \text{on}\quad \Gamma\times(0,...
- 121
2
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0
answers
133
views
How one could choose the value of viscous coefficient for obtaining stable solution of Burgers' equation?
Burgers' equation is a fundamental PDE used in various fields such as number theory, gas dynamics, heat conduction, elasticity, etc. It is crucial especially for developing numerical models for ...
- 125
2
votes
0
answers
193
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Modeling First Order Parabolic PDE (Battery Storage Model)
I'm trying to solve the following first order parabolic partial differential equation,
\begin{equation*}
X \frac{\partial V}{ \partial Q} = -\frac{1}{2} \sigma^2 \frac{\partial^2 V}{\partial X^2} + ...
- 215
2
votes
0
answers
822
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Solving a system of 4 coupled PDEs representing variable diffusivity
I have four partial differential equations representing mass conservation of two compressible fluid phases (marked by subscripts $p1$ and $p2$) in two different continuum media (marked by subscripts $...
user5510
2
votes
0
answers
155
views
Solvers for stiff initial value ODEs with sparse Jacobian
What ODE solvers are optimized for solving stiff systems with sparse Jacobian? Such systems appear, for instance, when a parabolic PDE is discretized in space using typical finite difference or finite ...
- 281
1
vote
0
answers
50
views
Effect of reducing flux consistency order at boundary on convergence order
Consider the 1D nonstationary convection-diffusion PDE
$$
\begin{alignat}{2}
\partial_t u &= -a \partial_x u + D \partial_{xx}u, &\qquad x \in (0,1), t \in (0,T), \\
f(t) &= \left.\left( a ...
- 401
1
vote
0
answers
78
views
Oscillations when solving parabolic heat equation with FTCS
I'm wondering if someone could help me out, or point me in a direction of how I can understand the following oscillations that occur when I solve the Porous Medium Equation $$u_t = u_{xx}^{m+1}$$ ...
- 111
1
vote
0
answers
63
views
Discretizing a parabolic PDE with finite volume method
I want to discretize the following parabolic PDE:
$$u_t = \nabla\cdot(\alpha(x)\nabla u)- \beta u\\
x\in\Omega \subset \mathbb{R}^2\\
\partial_n u = 0\\
u(t,0) = u_0(x)\ge 0, \alpha(x)>0$$
Given ...
- 216
1
vote
0
answers
310
views
Accuracy of finite difference method for heat equation on a disk
To study an approximation for the heat equation $$\frac{\partial^2 u}{\partial r^2}+\frac{1}{r}\frac{\partial u}{\partial r}+\frac{1}{r^2}\frac{\partial^2 u}{\partial\theta^2}=f(r,\theta)$$
on the ...
- 11
1
vote
0
answers
72
views
Growing error from a smooth initial condition for Fisher KPP equation
I'm studying the Fisker-KPP equation on the line (and in $]0, 100[$ numerically):
$$
\partial_t u = \Delta_{xx} u + u(1-u)
$$
I notice a behavior I don't understand with a smooth initial condition $...
- 433
1
vote
0
answers
90
views
Implementing initial conditions into the solution domain of a 1-D advection-diffusion equation
I have the following PDE.
$\frac{\partial c}{\partial t} + v\frac{\partial c}{\partial x} - D\frac{\partial^2 c}{\partial x^2} = 0
\\$.
I have discretized it such that i now have
$\frac{dC}{dt} = ...
- 11
1
vote
0
answers
45
views
explicit scheme stability restriction
Looking at the plain heat equation $u_t=u_{xx}$ the explicit scheme for it would look like the following iteration:
$$u_{m,n+1}=\rho u_{m-1,n}+(1-2\rho)u_{m,n}+\rho u_{m+1,n}$$
I noticed this equation ...
- 1,196
1
vote
0
answers
82
views
Stability of two PDEs
I have the following two PDEs which I want to check for stability:
$$u_t= u_{xx} , \ u(x,0)=1 , \ u_x(1,t)=-hu^4(1,t) , \ u_x(0,t) = 0 $$
$$u_t = u_{xx}-\sin(x+t)+\cos(x+t) , \ u(x,0)=\cos(x) ,\ u_x(...
- 181
1
vote
0
answers
165
views
Boundary Conditions for the given PDE
I'm working on the Black-Scholes equation, but I'm pretty new to financial modeling. Right now, I am trying to understand the Black-Scholes PDE. I understand that the Black-Scholes equation is given ...
- 215
1
vote
0
answers
1k
views
Finite Difference in Polar Co-ordinates
Background
Please note that I am duplicating the question on scicomp. I have already asked this in math.
I am trying to come up with a scheme in Polar Co-ordinates for the following PDE:
PDE I am ...
- 95
0
votes
0
answers
43
views
The physical meaning of conservative mass in diffusion equation
I am working on 1-D mass transient diffusion in a radial domain (spherical object) using finite volume method. My equation reads
$$\frac{\partial C}{\partial t} = D\left[\frac{\partial^2 C}{\partial r^...
- 1
0
votes
0
answers
85
views
fixed point iteration on DD method
I have to solve the the problem $u_t+\Delta^2u=f(u)$, where $f(u)$ is non-linear, using domain-decomposition method.
My approach is first using fixed point iteration on mixed form i.e to say $u^{k+1}...
- 41
0
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0
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62
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Solving Parabolic PDE using Matlab
I have the following pde (Burger's equation)
for $\epsilon>0: u_t+u.u_x=\epsilon.u_{xx}$ and $x\in \mathbb{R},t>0$
and the initial condition: $u(x,0)=\phi(x)=1_{(-\infty,0)}(x)$.
I want ...
- 109
0
votes
0
answers
83
views
Analytic vs discrete understanding of PDE
The PDE I am working with:
$$\partial_tu = \nabla \cdot (a(x)\nabla u)-\beta(x)u\\
\partial_nu=0, x \in \Omega \subset \mathbb{R}^2\\
\beta(x)>0$$
Integrate the PDE:
$$\int_\Omega \partial_t u=\...
- 216